Initial program 47.5
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
Initial simplification30.7
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\]
- Using strategy
rm Applied add-sqr-sqrt30.7
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}\]
Applied times-frac30.5
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]
Applied times-frac27.3
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}\]
Simplified27.3
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]
Simplified13.4
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \color{blue}{\left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\right)}\]
- Using strategy
rm Applied associate-*r*11.9
\[\leadsto \color{blue}{\left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}}\]
- Using strategy
rm Applied add-sqr-sqrt11.9
\[\leadsto \left(\frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\sqrt{\left|\frac{k}{t}\right|} \cdot \sqrt{\left|\frac{k}{t}\right|}}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\]
Applied div-inv11.9
\[\leadsto \left(\frac{\color{blue}{\frac{2}{t} \cdot \frac{1}{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|} \cdot \sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\]
Applied times-frac11.5
\[\leadsto \left(\color{blue}{\left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}}\right)} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\]
Applied associate-*l*11.2
\[\leadsto \color{blue}{\left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\right)} \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\]
- Using strategy
rm Applied *-un-lft-identity11.2
\[\leadsto \left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\right) \cdot \frac{\frac{\ell}{t}}{\color{blue}{1 \cdot \left|\frac{k}{t}\right|}}\]
Applied div-inv11.2
\[\leadsto \left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\right) \cdot \frac{\color{blue}{\ell \cdot \frac{1}{t}}}{1 \cdot \left|\frac{k}{t}\right|}\]
Applied times-frac11.3
\[\leadsto \left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\right) \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\frac{1}{t}}{\left|\frac{k}{t}\right|}\right)}\]
Simplified11.3
\[\leadsto \left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\right) \cdot \left(\color{blue}{\ell} \cdot \frac{\frac{1}{t}}{\left|\frac{k}{t}\right|}\right)\]
Final simplification11.3
\[\leadsto \left(\ell \cdot \frac{\frac{1}{t}}{\left|\frac{k}{t}\right|}\right) \cdot \left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\right)\]
